Author

Ji Hoon Ryoo

Date of Award

2001

Level of Access

Open-Access Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics

Advisor

David M. Bradley

Second Committee Member

Henrik Bresinsky

Third Committee Member

William Snyder

Abstract

At the beginning of my research, I understood the shuffle operation and iterated integrals to make a new proof-method (called a combinatorial method). As a first work, I proved an combinatorial identity 2 using a combinatorial method. While proving it, I got four identities and showed that one of them is equal to an analytic identity 1 which is found at the paper [2] written by David M. Bradley and Doug Bowman. Furthermore, I derived an formula involving nested harmonic sums. Using Maple (a mathematical software), I found a new combinatorial identity 3 and derived two formulas: One is related to multiple polylogarithms and the other is related to rational functions. Since letters in the identities represent differential 1-forms which converge, I can find new formulas - - - if I get a proper setting. - My research was developed by considering a combinatorial identity 4 given by David M. Bradley, thesis advisor. Though it looked very complicated, the implication for the identity was very interesting to me. Using a combinatorial proof-method, I proved it. Even though I just derived one formula involving nested harmonic sums in this thesis, the identity has potentiality because, if I 6nd a new setting for differential 1-forms, I can derive a new formula involving multiple polylogarithms. It was not very easy to prove the combinatorial identity 4 even though I used the combinatorial proof-method as I did at the proofs of the combinatorial identity 2 and 3. The reason is that the result of the identity 4 is more complicated than those of the identities 3 and 4. So, Lemma 5 is needed to complete the proof of the identity 4, which step is not needed in the proofs of the identities 3 and 4. When formulating the identity 4, I had a trouble in defining the notations because of their complexity. When I formulated the identity 4, it was a beautiful formula. As we can see in the paper [3], there are various conjectures related to multiple zeta values whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. At this situation, this combinatorial proof-method can play a crucial role in developing other fields such as knot theory and quantum field theory as well as combinatorics.

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